Originally Posted by robherc
...If you figure that the amount of propulsion of any propeller results from its ability to "push" air towards the opposite direction as the propeller is attempting to travel, it makes this easier....
May I suggest it's helpful to think of a helicopter analogy. There are three different situations the rotor can be in. When the helicopter is climbing or in a level hover, the air moves downward through the rotor disk and is accelerated by the rotor, producing the lift that supports the helicopter's weight.
When the helicopter is descending rapidly, the air is passing through the rotor disk from bottom to top. However, the rotor decelerates the air, so that again the force is in the upward direction.
In between is the vortex ring state, obtained when the helicopter is descending moderately slowly. The air moves up through the rotor disk in the center, and down through the rotor disk toward the edge. Accompanied by turbulence and ineffective production of lift. But lifting nonetheless.
The Bauer machine just starting is like the autorotating helicopter. The air is moving from the back of the rotor to the front. The drag moves the machine forward, and the wheels apply torque to the rotor. However, there may be a positive aerodynamic torque from the autorotating rotor as well. If the torque from the wheels is greater than the aerodynamic torque, then the wheels can be said to be powering the rotor. If the aerodynamic torque is greater than the torque from the wheels, then the rotor may actually be powering the wheels. But the difference is really not very significant because the rotor is still producing forward thrust in either case. The aerodynamic torque comes from the fact that the lift vector is tilted forward relative to the blade chord, just like that of a glider whose descending trajectory tilts its lift vector forward and opposes the horizontal component of its drag.
As the Bauer machine picks up speed, it is like a helicopter pulling out of a vertical autorotating descent. Power is applied to the rotor and its velocity slows relative to the air, and then reverses. The vortex ring state forms and then shrinks toward the hub as more of the air flows from front to back through the rotor, until the air is flowing from front to back all along the rotor disk. Initially, this will be due to the induced velocity from the rotor's thrust as the machine has not quite yet reached the freestream wind velocity. Some parts of the inner disk may be producing an autorotating torque, but the outer portions are not, requiring the application of external power to keep the rotor turning. The wheels are powering the rotor by this point and the rotor can no longer be viewed as a windmill, if it ever was.
But as the machine continues to accelerate, it reaches and exceeds the velocity of the air, like a helicopter that finally starts to gain altitude. Now the velocity through the rotor disk is the sum of the machine's relative wind and the induced velocity. There is no doubt at this point that it is the wheels that are powering the rotor and not the other way around, because the lift vector, which is perpendicular to the vector sum of the rotational speed and the inflow, is tilted back and has a component in the plane of the disk that would retard the blade if it were not for the torque being applied at the hub.
Throughout, thrust from the rotor keeps the machine moving forward. Only in the beginning can the torque to the wheels be adding to the rotor thrust. As the craft accelerates, the wheel torque lessens and then reverses, acting as a drag on the vehicle. But as long as the thrust is greater than the sum of the aerodynamic drag and the drag from the wheels, the craft will accelerate. Eventually, the velocity through the rotor disk will become so high that the lift vector will be tilted back too far and the aerodynamic drag of the rotor will acquire a backward (upwind) component, and the thrust will drop off. When the thrust matches the drag from the wheels, the craft will have achieved steady-state operation.
Power is a tricky thing to work out for any sailing craft, because the power available changes. Power is thrust times velocity, so at rest the thrust horsepower is zero, even though the thrust is not. As the yacht accelerates, the thrust horsepower increases because the thrust increases as the rotor becomes more efficient (especially if the pitch is fixed), and because the velocity is increasing. As the thrust starts to drop, power still increases, but more slowly. (Just like the peak power occurs at a higher rpm than the peak torque of your car's engine.) But the power extracted by the wheels is also increasing because of both the increasing torque demands of the rotor and the higher speed of the machine. Eventually, the two curves come together, and the power balance is achieved at steady state. Either the energy flows (power) or force & moment sums can be used to calculate what's happening, but as a practical matter, it's easier to use the forces & moments.
So while a landyacht is like a fixed wing glider soaring in a thermal, the Bauer machine is like a helicopter, powered by the transmission from the wheels.